162 CHAPTER 5. BROWNIAN MOTION
Section Starter Question
Some mathematical objects are defined by a formula or an expression. Some
other mathematical objects are defined by their properties, not explicitly by
an expression. That is, the objects are defined by how theyact, not by what
theyare. Can you name a mathematical object defined by its properties?
Key Concepts
- We define Brownian motion in terms of the normal distribution of the
increments, the independence of the increments, and the value at 0. - The joint density function for the value of Brownian motion at several
times is a multivariate normal distribution.
Vocabulary
- Brownian Motionis the physical phenomenon named after the En-
glish botanist Robert Brown who discovered it in 1827. Brownian mo-
tion is the zig-zagging motion exhibited by a small particle, such as a
grain of pollen, immersed in a liquid or a gas. The first explanation
of this phenomenon was given by Albert Einstein in 1905. He showed
that Brownian motion could be explained by assuming the immersed
particle was constantly buffeted by the actions of the molecules of the
surrounding medium. Since then the abstracted process has been used
beneficially in such areas as analyzing price levels in the stock market
and in quantum mechanics. - TheWiener processis the mathematical definition and abstraction of
the physical process as a stochastic process. The American mathemati-
cian Norbert Wiener gave the definition and properties in a series of
papers starting in 1918. Generally, the termsBrownian motionand
Wiener processare the same, although Brownian motion emphasizes
the physical aspects and Wiener process emphasizes the mathematical
aspects. - Bachelier processis an uncommonly applied term meaning the same
thing as Brownian motion and Wiener process. In 1900, Louis Bachelier
introduced the limit of random walk as a model for the prices on the